A swimming pool can be filled by any of three hoses A, B or C. Hoses A and B together take 4 hours to fill the pool. Hoses A and C together take 5 hours to fill the pool. Hoses B and C together take 6 hours to fill the pool. How many hours does it take hoses A, B and C working together to fill the pool? Express your answer as a decimal to the nearest hundredth.
Explanation: Let the rate that hose $A$ fills the pool be equal to $A$, and similarly for hoses $B$ and $C$.  Then, let $P$ be equal to the volume of the pool.  From the given information, we can write the equation $P=4(A+B)$, which just says the pool volume is equal to the rate it is being filled, multiplied by the time taken to fill it.  We can rewrite this as $\frac{P}{4}=A+B$.  Doing this with the rest of the given information, we can write three equations: $$\frac{P}{4}=A+B$$ $$\frac{P}{5}=A+C$$ $$\frac{P}{6}=B+C$$ Adding these three equations, we can simplify as shown: \begin{align*}
\frac{P}{4}+\frac{P}{5}+\frac{P}{6}&=(A+B)+(A+C)+(B+C)\\
\Rightarrow\qquad \frac{15P}{60}+\frac{12P}{60}+\frac{10P}{60}&=2(A+B+C)\\
\Rightarrow\qquad 37P&=120(A+B+C)\\
\Rightarrow\qquad P&=\frac{120}{37}(A+B+C)
\end{align*} Looking carefully at the final expression here, we can see that $A+B+C$ is the rate that the pool would be filled with all three hoses working together.  So, $\frac{120}{37}\approx \boxed{3.24}$ is equal to the number of hours it would take all three hoses to fill the pool.